SUMMARY

The total force on a charge \(q\) moving with velocity \(v\) in the presence of magnetic and electric fields \(\mathbf{B\) and \(\mathbf{E\), respectively is called the Lorentz force. It is given by the expression: \[ \mathbf{F}=q(\mathbf{v} \times \mathbf{B}+\mathbf{E}) \] The magnetic force \(q(\mathbf{v} \times \mathbf{B})\) is normal to \(\mathbf{v\) and work done by it is zero.
A straight conductor of length \(l\) and carrying a steady current \(I\) experiences a force \(\mathbf{F}\) in a uniform external magnetic field \(\mathbf{B}\), \[ \mathbf{F}=I \mathbf{1} \times \mathbf{B} \] where \(|\mathbf{l}| = l\) and the direction of \(\mathbf{l}\) is given by the direction of the current.
In a uniform magnetic field \(\mathbf{B}\), a charge q executes a circular orbit in a plane normal to \(\mathbf{B}\). Its frequency of uniform circular motion is called the cyclotron frequency and is given by: \[ v_{c}=\frac{q B}{2 \pi m} \] This frequency is independent of the particle’s speed and radius. This fact is exploited in a machine, the cyclotron, which is used to accelerate charged particles.
The Biot-Savart law asserts that the magnetic field \(d\mathbf{B}\) due to an element \(d\mathbf{l}\) carrying a steady current \(I\) at a point \(P\) at a distance \(r\) from the current element is: \[ \mathrm{d} \mathbf{B}=\frac{\mu_{0}}{4 \pi} I \frac{\mathrm{d} \mathbf{l} \times \mathbf{r}}{r^{3}} \] To obtain the total field at \(P\), we must integrate this vector expression over the entire length of the conductor.
The magnitude of the magnetic field due to a circular coil of radius \(R\) carrying a current \(I\) at an axial distance \(x\) from the centre is \[ B=\frac{\mu_{0} I R^{2}}{2\left(x^{2}+R^{2}\right)^{3 / 2}} \] At the centre this reduces to \[ B=\frac{\mu_{0} I}{2 R} \]
Ampere’s Circuital Law: Let an open surface \(S\) be bounded by a loop \(C\). Then the Ampere’s law states that \(\oint_{C} \mathbf{B} \cdot d \mathbf{l}=\mu_{0} I\) where \(I\) refers to the current passing through \(S\). The sign of \(I\) is determined from the right-hand rule. We have discussed a simplified form of this law. If \(\mathbf{B}\) is directed along the tangent to every point on the perimeter \(L\) of a closed curve and is constant in magnitude along perimeter then, \[ BL=\mu_0 I_e \] where \(I_e\) is the net current enclosed by the closed circuit.
The magnitude of the magnetic field at a distance \(R\) from a long, straight wire carrying a current \(I\) is given by: \[ B=\frac{\mu_{0} I}{2 \pi R} \] The field lines are circles concentric with the wire.
The magnitude of the field \(B\) inside a long solenoid carrying a current \(I\) is \[ B=\mu_0nI \] where \(n\) is the number of turns per unit length. For a toroid one obtains, \[ B=\frac{\mu_{0} N I}{2 \pi r} \] where \(N\) is the total number of turns and \(r\) is the average radius.
Parallel currents attract and anti-parallel currents repel.
A planar loop carrying a current \(I\), having \(N\) closely wound turns, and an area \(A\) possesses a magnetic moment \(\mathbf{m}\) where, \[ \mathbf{m}=\mathrm{N} I \mathbf{A} \] and the direction of \(\mathbf{m}\) is given by the right-hand thumb rule : curl the palm of your right hand along the loop with the fingers pointing in the direction of the current. The thumb sticking out gives the direction of \(\mathbf{m}\) (and \(\mathbf{A}\))
When this loop is placed in a uniform magnetic field \(\mathbf{B}\), the force \(\mathbf{F}\) on it is: \(F = 0\)
And the torque on it is, \[ \tau=\mathbf{m} \times \mathbf{B} \] In a moving coil galvanometer, this torque is balanced by a countertorque due to a spring, yielding \[ k \phi=N I A B \] where \(\phi\) is the equilibrium deflection and \(k\) the torsion constant of the spring.
An electron moving around the central nucleus has a magnetic moment \(\mu_l\) given by: \[ \mu_{l}=\frac{e}{2 m} l \] where \(l\) is the magnitude of the angular momentum of the circulating electron about the central nucleus. The smallest value of \(\mu_l\) is called the Bohr magneton \(\mu_B\) and it is \(\mu_B= 9.27×10^{–24} J/T\)
A moving coil galvanometer can be converted into a ammeter by introducing a shunt resistance \(r_s\), of small value in parallel. It can be converted into a voltmeter by introducing a resistance of a large value in series.

Physical Quantity	Symbol	Nature	Dimesnions	Units	Remarks
Permeability of free space	\( \mu_0 \)	Scalar	\( [MLT^{-2}A^{-2}] \)	\( TmA^{-1} \)	\(4\pi\times 10^{-7} TmA^{-1}\)
Magnetic Field	\( \mathbf{B} \)	Vector	\( [MT^{-2}A^{-1}] \)]	\( T \)(tesla)
Magnetic Moment	\( \mathbf{m} \)	Vector	\( [L^2A] \)]	\( Am^2 \) or \(J/T\)
Torsion Constant	\( k \)	Scalar	\( [ML^2T^{-2}] \)	\( Nm\text{ rad}^{-1} \)	Appears in MCG

POINTS TO PONDER

Electrostatic field lines originate at a positive charge and terminate at a negative charge or fade at infinity. Magnetic field lines always form closed loops.
The discussion in this Chapter holds only for steady currents which do not vary with time.
When currents vary with time Newton’s third law is valid only if momentum carried by the electromagnetic field is taken into account.
Recall the expression for the Lorentz force, \[ \mathbf{F}=q(\mathbf{v} \times \mathbf{B}+\mathbf{E}) \] This velocity dependent force has occupied the attention of some of the greatest scientific thinkers. If one switches to a frame with instantaneous velocity \(\mathbf{v}\), the magnetic part of the force vanishes. The motion of the charged particle is then explained by arguing that there exists an appropriate electric field in the new frame. We shall not discuss the details of this mechanism. However, we stress that the resolution of this paradox implies that electricity and magnetism are linked phenomena (electromagnetism) and that the Lorentz force expression does not imply a universal preferred frame of reference in nature.
Ampere’s Circuital law is not independent of the Biot-Savart law. It can be derived from the Biot-Savart law. Its relationship to the Biot-Savart law is similar to the relationship between Gauss’s law and Coulomb’s law.